LESSONPLAN

MATHEMATICS – KS4

WHY TEACHTHIS?

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CROSSINGLINES

Visualisation and spatialreasoning are importantskills that enable us to makesense of geometricalsituations and predict howpoints and lines will behave.This lesson gives studentsthe opportunity to examinepositions that are the same distance from twofixed points.

Geometrical constructions suchas perpendicular bisectors andangle bisectors are often given tostudents as procedures to follow,and students don’t alwaysunderstand why the proceduresthat they have been taughtproduce the construction of thatname. For example, why doesthe procedure for drawing aperpendicular bisector of a linesegment actually bisect it? Andwhy does it do so at right angles?In this lesson, students areencouraged to think about whythis construction works and howthey might justify it. They areprovided with ample opportunityto practise the technique on acoordinate grid and improvetheir precision while alsoexploring the points throughwhich their perpendicularbisectors pass. Attempting tofind line segments whoseperpendicular bisectors passthrough or near to the origingives students a reason to beaccurate in their constructions.They have to think carefullyabout the relationship betweenthe ends of the line segment andthe direction of the perpendicularbisector in order to anticipatewhere the perpendicularbisectors of their chosen linesegments will go.

STARTER ACTIVITY

THE REASONING BEHIND GEOMETRICAL CONSTRUCTIONS OFFERSOPPORTUNITIES FOR STUDENTS TO DEVELOP THEIR

UNDERSTANDING OF POINTS AND LINES, SAYS COLIN FOSTER...

Q. I want you to imagine twoidentical circles next to each other,one on the left and one on the right. (Close your eyes if that helps.) Now imagine that the circles slowlymove closer together until theyoverlap. Can you describe whatyou see?

The circles overlap at two positions.A student could draw on the boardwhat they see, or you could draw orproject this image on the board:

Q. I’m going to join up the centresof the two circles with a purple line.

Q. Now, in red, I’m going to join up the two points where the circles intersect.

Q. What can you say about the red line?

Students will probably say that isvertical, or at right angles(perpendicular) to the horizontal line.

Q. How do you know that the purpleand red lines are perpendicular?

This is quite hard to justifyrigorously, without vague referencesto “symmetry”, and students coulddiscuss this in pairs. To see why theangles created must be right anglescould involve drawing in someadditional lines, such as the bluelines below, creating the bluerhombus (it is a rhombus since all ofthe sides are equal to the radius ofthe circles).

Now we have two pairs ofcongruent isosceles triangles(shown below): one pair with thepurple length as the third side, andone pair with the red length as thethird side. These isosceles triangles

mean that we have pairs of equalangles, shown shaded below.

So in the original diagram thismeans that the four black angles inthe centre must be equal to eachother, and since they add up to360° each one must be 90°.Students are likely to need helpputting these ideas together tomake a convincing argument.

This means that two overlappingcircles of the same size gives us agood way to construct right angles.Students will probably have seenthis perpendicular bisectorconstruction before, but may nothave thought very carefully aboutwhy it works.

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LESSONPLAN MATHEMATICS | KS4

STRETCH THEM FURTHERCONFIDENT STUDENTS COULD CHOOSETHREEPOINTS AND CONSTRUCT THEPERPENDICULAR BISECTORS OF THE LINESEGMENTS JOINING EACH PAIR OF POINTS.PROVIDED THAT THESE POINTS ARE NOTCOLLINEAR (LIE ON A STRAIGHT LINE), THEYWILL FORM A TRIANGLE, AND THEPERPENDICULAR BISECTORS OF A TRIANGLEARE COINCIDENT – I.E., THEY ALL PASSTHROUGH THE SAME POINT. THIS POINT ISCALLED THE CIRCUMCENTREOF THETRIANGLE, AND A CIRCLE (THECIRCUMCIRCLE) CAN BE DRAWN, WITH THISPOINT AS ITS CENTRE, PASSING THROUGH ALLTHREE VERTICES OF THE TRIANGLE. STUDENTSCOULD TRY TO FIND A TRIANGLE WHOSECIRCUMCENTRE IS THE ORIGIN. (ONE EXAMPLEWOULD BE THE TRIANGLE WITH VERTICES AT (-1, 3), (3, -1) AND (1, -3).)

INFORMATIONCORNER

Colin Foster is an AssistantProfessor in mathematicseducation in the School ofEducation at the Universityof Nottingham. He haswritten many books and articles for mathematics teachers (see www.foster77.co.uk).

ABOUT OUR EXPERT

ADDITIONAL RESOURCESA NATURAL ENVIRONMENT FOR EXPLORINGPERPENDICULAR BISECTORS IS THATPROVIDED BY GEOGEBRA, WHICH ISAVAILABLE FREE OF CHARGE FROMWWW.GEOGEBRA.ORG/CMS/EN. IT IS EASY TO GENERATE A PERPENDICULARBISECTOR OF A LINE SEGMENT AND THENMOVE THE ENDS OF THE LINE SEGMENTAROUND AND SEE WHAT HAPPENS TO THEPERPENDICULAR BISECTOR.

You could conclude the lesson with a plenary in which studentstalk about the examples that they have found and what they

have learned. They should have had ample opportunity to practise their skills inconstructing perpendicular bisectors. They might also have become more adept atanticipating where such lines will go.

The five lines given all pass quite close to the origin. The order of the lines, in terms of howclose they get to the origin, is: 5 < 2 < 4 < 1 < 3. Line 5 passes through the origin exactly. The approximate distances are given in the table below.

What other points did students choose and how close did the associated perpendicularbisectors get to the origin? To check students’ work it may be useful to know that ingeneral the closest distance from the origin to the perpendicular bisector of the linesegment joining (a, b) to (c, d) (assuming that these two points are di!erent) is given by:

So for the example in which the line segment went from (–3, 3) to (1, 4) this gives a distance of 0.12 (correct to 2 decimal places).

DISCUSSION

MAIN ACTIVITYPut some axes on the board from –5 to 5 both ways.

Q. Can someone tell me whereto plot the points (–1, 2) and (3, 4),please.

Students could come to theboard to show where the pointsare. If no mistakes are made, youcould ask students whatmistakes someone might make,if you think that these could arisein the class. For example,coordinates might be plotted thewrong way round, such as (4, 3)instead of (3, 4), or negativenumbers might cause confusion,so students might plot (1, 2)instead of (–1, 2), for example.

Q. Let’s label (–1, 2) as A and (3, 4) as B. Can you tell me apoint that is exactly the samedistance from A as it is from B?

Students will probably say (1, 3),and you can put a cross there.

Q. And another? And another?

This is harder now. Place a crossat whatever points students say,unless other students object, inwhich case ask them to say whythey object. Carry on until youhave 4 or 5 points. Because of

the visual e"ect of the axes, somestudents may think that pointssuch as (0, 5) are nearer to A thanto B. Students could measure orcount squares to decide whetherthey agree or not. Students maysuggest points that don’t haveinteger coordinates, which is fine.

Q. What’s happening? Why?

Try to get students to describewhat they see in words. A line isbeing formed that is at right anglesto a line joining A and B. Theymight use words likeperpendicular bisector – if they

don’t, then you could – or theymight try to find an equation for the new line (e.g., 2x + y = 5 or y = –2x + 5).

Students may not immediately seehow this relates back to the starter.If they carry out the constructionfor a perpendicular bisector of ABthey will get the line that passesthrough the points marked with across. See if they can explain whythis is.

Q. Draw some more axes from –5to 5 both ways and plot the points(–3, 3) and (1, 4), please. Constructthe perpendicular bisector. Does itgo through the origin?

This is a good test of accuracy,because the perpendicularbisector goes near to the originbut doesn’t pass through it exactly.

Q. I want you to construct theperpendicular bisectors to theseline segments and decide whichone passes closest to the origin:

Line 1: (0, 2) to (2, 1)Line 2: (0, 3) to (3, –1)

Line 3: (–1, 2) to (2, –2)Line 4: (–3, –4) to (2, 5)Line 5: (–3, –2) to (2, 3)

You need to do it very accuratelyto be sure!What other pairs of points can you find that have perpendicularbisectors that pass near to the origin?

To estimate the closest distance ofthe perpendicular bisectors fromthe origin, students could measureby eye with a ruler or use a setsquare to help. Another way wouldbe to make an accurate drawingon a large piece of paper and thenuse compasses to construct aperpendicular line from the originto the perpendicular bisector, andthen measure the length of thisline. (If a perpendicular bisectorpasses very close to the origin, thiswill be a bit fiddly, but it will beeasier with lines that do notapproach the origin so closely.)Students could make a poster toillustrate their “best” examples.

A

X

X

X

X

B

0

1

2

3

4

5

6

0-1 1 2 3 4

0 1

1

-1

-1

-2

-2

-3 2

2

3

4

5

6

00

Line Closest distance to origin (correct to 2 decimal places)

1 0.22

2 0.10

3 0.30

4 0.19

5 0.00

a| |+2 2 2 2b c d–

–

–

2 (a )c 2+ –(b d 2)